We firstly prove that -times integrated -resolvent operator function ( -ROF) satisfies a functional equation which extends that of -times integrated semigroup and -resolvent operator function. Secondly, for the inhomogeneous -Cauchy problem , , , if is the generator of an -ROF, we give the relation between the function and mild solution and classical solution of it. Finally, for the problem , , , where is a linear closed operator. We show that generates an exponentially bounded -ROF on a Banach space if and only if the problem has a unique exponentially bounded classical solution and Our results extend and generalize some related results in the literature. 1. Introduction This paper is concerned with the properties of -integrated -resolvent operator function ( -ROF) and two inhomogeneous fractional Cauchy problems. Throughout this paper, , denotes the set of natural numbers. . Let be Banach spaces, denote the space of all bounded linear operators from to , . If is a closed linear operator, denotes the resolvent set of and denotes the resolvent operator of . denotes the space of -valued Bochner integrable functions: with the norm , it is a Banach space. By we denote the convolution of functions denotes the function and , the Dirac delta function. In 1997, Mijatovi？ et al.  introduced the concept of -times integrated semigroup which extends -times integrated semigroup , they showed an to be the pseudoresolvent of a -times integrated semigroup if and only if satisfies the following functional equation: In the special case of , the corresponding result is summarized in . For the inhomogeneous Cauchy problem where , , , and is the generator of a -times integrated semigroup on a Banach space for some . Let . Lemmas and of  show that if there is a mild(classical) solution of (4), then and . On the other hand, if , then is also a mild (classical) solution of it. Furthermore, if generates an exponential bounded -times integrated semigroup on a Banach space , then, for any , is the unique exponential bounded classical solution of the following problem: In recent years, a considerable interest has been paid to fractional evolution equation due to its applications in different areas such as stochastic, finance, and physics; see [3–8]. One of the most important tools in the theory of fractional evolution equation is the solution operator (fractional resolvent family) [9–15]. The notion of solution operator was developed to study some abstract Volterra integral equations  and was first used by Bajlekova  to study a class of fractional order
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.